To explore the high-frequency radio spectra of galaxies in clusters, we used NRAO's Very Large Array at four frequencies, 4.9-43 GHz, to observe 139 galaxies in low redshift (z < 0.25), X-ray detected, clusters. The clusters were selected from the survey conducted by Ledlow and Owen, who provided redshifts and 1.4 GHz flux densities for all the radio sources. We find that more than half of the observed sources have steep microwave spectra as generally expected (α < –0.5, in the convention S vprop να). However, 60%-70% of the unresolved or barely resolved sources have flat or inverted spectra. Most of these show an upward turn in flux at ν>22 GHz, implying a higher flux than would be expected from an extrapolation of the lower-frequency flux measurements. Our results quantify the need for careful source subtraction in increasingly sensitive measurements of the Sunyaev-Zel'dovich effect in clusters of galaxies (as currently being conducted by, for instance, the Atacama Cosmology Telescope and South Pole Telescope groups).